# P3036: Erdős's Problem on Affine Copies of Infinite Sets in Sets of Positive Measure

- ID: `P3036`
- Reference: `erdos-problem-120`
- Page: https://theoremdb.org/statements/P3036
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 120: For a set $A \subseteq \mathbb{R}$, say that a set $E \subseteq \mathbb{R}$ avoids affine copies of $A$ if there do not exist $a, b \in \mathbb{R}$ with $a \neq 0$ such that $\{a \cdot x + b : x \in A\} \subseteq E$. Let $A \subseteq \mathbb{R}$ be an infinite set. Must there exist a measurable set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that avoids affine copies of $A$?

### Context

This problem belongs to the intersection of combinatorial geometry and measure theory, asking whether infinite subsets of the real line can always be excluded from some set of positive measure under all non-degenerate affine transformations.

### Problem setup

- **Definition (Definition 1).** For a set $A \subseteq \mathbb{R}$, a set $E \subseteq \mathbb{R}$ avoids affine copies of $A$ if there do not exist real numbers $a$ and $b$ with $a \neq 0$ such that the image of $A$ under the map $x \mapsto a \cdot x + b$ is contained in $E$.
- **Definition (The Lebesgue measure of a measurable set $E \subseteq \mathbb{R}$).** The Lebesgue measure of a measurable set $E \subseteq \mathbb{R}$ is denoted $\text{volume}(E)$; the set has positive measure if $0 < \text{volume}(E)$.
- **Remark.** This problem belongs to the intersection of combinatorial geometry and measure theory, asking whether infinite subsets of the real line can always be excluded from some set of positive measure under all non-degenerate affine transformations.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 120: For a set $A \subseteq \mathbb{R}$, say that a set $E \subseteq \mathbb{R}$ avoids affine copies of $A$ if there do not exist $a, b \in \mathbb{R}$ with $a \neq 0$ such that $\{a \cdot x + b : x \in A\} \subseteq E$. Let $A \subseteq \mathbb{R}$ be an infinite set. Must there exist a measurable set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that avoids affine copies of $A$?

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 120 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 120: For a set $A \subseteq \mathbb{R}$, say that a set $E \subseteq \mathbb{R}$ avoids affine copies of $A$ if there do not exist $a, b \in \mathbb{R}$ with $a \neq 0$ such that $\{a \cdot x + b : x \in A\} \subseteq E$. Let $A \subseteq \mathbb{R}$ be an infinite set. Must there exist a measurable set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that avoids affine copies of $A$? [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Current status and unresolved remainder).** OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 120 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 120: For a set $A \subseteq \mathbb{R}$, say that a set $E \subseteq \mathbb{R}$ avoids affine copies of $A$ if there do not exist $a, b \in \mathbb{R}$ with $a \neq 0$ such that $\{a \cdot x + b : x \in A\} \subseteq E$. Let $A \subseteq \mathbb{R}$ be an infinite set. Must there exist a measurable set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that avoids affine copies of $A$?

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 120 as open. The unresolved remainder is the full displayed statement.

A complete resolution must satisfy this condition: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 120: For a set $A \subseteq \mathbb{R}$, say that a set $E \subseteq \mathbb{R}$ avoids affine copies of $A$ if there do not exist $a, b \in \mathbb{R}$ with $a \neq 0$ such that $\{a \cdot x + b : x \in A\} \subseteq E$. Let $A \subseteq \mathbb{R}$ be an infinite set. Must there exist a measurable set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that avoids affine copies of $A$?

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 120 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 120 was open at commit 8138974387d9030542daabe67faaa33eff9356f8.
- The pinned Formal Conjectures declaration was matched by problem number and source locator.
- The controlled TheoremDB source corpus was checked for an already published record with the same slug.

### Open directions

- **Route 1** (reported): Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 120: For a set $A \subseteq \mathbb{R}$, say that a set $E \subseteq \mathbb{R}$ avoids affine copies of $A$ if there do not exist $a, b \in \mathbb{R}$ with $a \neq 0$ such that $\{a \cdot x + b : x \in A\} \subseteq E$. Let $A \subseteq \mathbb{R}$ be an infinite set. Must there exist a measurable set $E \subseteq \mathbb{R}$ of positive Lebesgue measure that avoids affine copies of $A$? [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-120`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Erdős Problems database, Problem 120, maintained status record. Erdős Problems record 120, checked 2026-08-01. Problem 120; status field and linked bibliography https://www.erdosproblems.com/120
   - Also cited at Problem 120; status snapshot 8138974387d9030542daabe67faaa33eff9356f8
   - Also cited at See dataset.references[0] for the exact external source and locator.
   - Also cited at Editorial research route recorded 2026-07-31
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source use: original_summary
   - Records the current open status and links the literature attached to this exact numbered problem.
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - For Erdős's Problem on Affine Copies of Infinite Sets in Sets of Positive Measure: Records the current open status and links the literature attached to this exact numbered problem.
   - Source named by the research packet.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 120 entry in data/problems.yaml https://github.com/teorth/erdosproblems/blob/8138974387d9030542daabe67faaa33eff9356f8/data/problems.yaml
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source use: original_summary
   - Pins the maintained database snapshot used for this release's dated status decision.
   - Source used to assess the problem's recorded status.
   - For Erdős's Problem on Affine Copies of Infinite Sets in Sets of Positive Measure: Pins the maintained database snapshot used for this release's dated status decision.
3. <a id="reference-3"></a>Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 120. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/120.lean:L45; theorem erdos_120; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/120.lean#L45
   - reference_database; primary source; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1; checked 2026-07-31
   - Source use: original_summary
   - Supplies the pinned formal declaration whose human-readable editorial statement is published here.
   - Source used to assess the problem's recorded status.
   - For Erdős's Problem on Affine Copies of Infinite Sets in Sets of Positive Measure: Supplies the pinned formal declaration whose human-readable editorial statement is published here.